Skip to main content

Estimates of the Solutions of a System of Quasi-linear PDEs. A Probabilistic Scheme

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1832)

Abstract

We propose a probabilistic scheme to estimate the Hölder norm and the gradient of the solutions of a system of quasi-linear PDEs of parabolic type. Indeed, thanks to the theory of Forward Backward stochastic differential equations, we are able to give a stochastic representation of the solutions of such systems of PDEs. Making use of Krylov and Safonov estimates, we deduce a Hölder estimate of these solutions in the case of uniformly parabolic systems with measurable coefficients. Moreover, from a variant of the Malliavin–Bismut integration by parts formula, we establish under appropriate assumptions an estimate of the supremum norm of the gradient of these solutions.

Keywords: Forward-backward stochastic differential equation, gradient estimate, Hölder estimate, integration by parts, system of quasi-linear PDEs of parabolic type.

Keywords

  • Parabolic System
  • Parabolic Type
  • Probabilistic Scheme
  • Part Formula
  • Heat Semigroup

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Delarue, F. (2003). Estimates of the Solutions of a System of Quasi-linear PDEs. A Probabilistic Scheme. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXVII. Lecture Notes in Mathematics, vol 1832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-40004-2_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-40004-2_12

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20520-3

  • Online ISBN: 978-3-540-40004-2

  • eBook Packages: Springer Book Archive